C₅×#63805 #13087077

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#63805

Splittings

Hall	C₃²×C₅ ⋊ #131
Sylow	C₃² ⋊ C₅×#131

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₄×C₅
Frattini	C₄	C₂×C₅×D₃²
Fitting	C₃²×C₅×C₈	C₂²
socle	C₂×C₃²×C₅	C₂²×C₄
prime radical	C₅×#3798	C₂
cosocle	C₃²×C₄	C₂³×C₅
hypercenter	C₅×C₈	D₃²
hypocenter	C₃²	C₅×#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄×C₅ ◁ C₅×C₈

270 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3,4,5)(6,8,9,16,10,15,17,21)(7,12,13,19,11,14,18,20)(22,23,24)(25,26,27);(7,11)(8,15)(13,18)(16,21)(23,24);(6,7)(8,14)(9,13)(10,11)(12,15)(16,20)(17,18)(19,21)(26,27)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 3394);